Let $\xi_1, \xi_2, \cdots$ be independent random variables. The distribution of $\max (0, \xi_1, \xi_1 + \xi_2, \cdots, \xi_1 + \cdots + \xi_n)$ is investigated by means of a method based on the construction of certain events with easily determined proabilities. These yield a new formula for the distribution of the maximum which is sometimes more useful than that given in literature.
"On the Distribution of the Maximum of the Sequence of Sums of Independent Random Variables." Ann. Probab. 3 (2) 289 - 297, April, 1975. https://doi.org/10.1214/aop/1176996399